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An Improved Exponential Ratio-Type Estimator with Two Auxiliary
Variables in Double Sampling with Nonresponse
Faweya, Olanrewaju
1
, Abifade, Victor Oluwatobi
2*
, Akinyemi, Oluwadare
3
, Oyinloye, Adedeji Adigun
4
,
Ajayi, Esther Dunsin
5
, Oniyinde, Yetunde Omolara
6
1
Department of Statistics, Ekiti State University, Ado Ekiti, Nigeria.
2 ,4,6
Department of Mathematical Sciences, Bamidele Olumilua University of Education, Science and
Technology, Ikere Ekiti, Nigeria.
3
Department of Statistics, Ekiti State University, Ado Ekiti, Nigeria.
5
Department of Statistics, Ekiti State University, Ado Ekiti, Nigeria.
*Corresponding Author
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150300035
Received: 16 March 2026; Accepted: 21 March 2026; Published: 07 April 2026
ABSTRACT
This research introduces a new exponential ratio-type estimator for estimating the population mean when survey
data are affected by nonresponse. The proposed estimator utilizes two auxiliary variables within a double
sampling framework in order to enhance estimation accuracy. Expressions for the mean squared error (MSE),
coefficient of variation (CV), and relative efficiency (RE) of the estimator were derived. The performance of the
estimator was assessed using numerical illustrations and compared with some existing estimators. The empirical
results reveal that the proposed estimator yields smaller MSE and CV values while achieving higher relative
efficiency across different subsampling rates at a 5% nonresponse level. The findings indicate that the integration
of auxiliary variables through exponential adjustments leads to substantial improvement in estimator
performance. Consequently, the proposed estimator provides a more reliable and efficient alternative for
estimating population means in the presence of nonresponse.
Keywords: Exponential ratio-type estimator, Nonresponse, Mean square error, Relative efficiency, Auxiliary
information.
INTRODUCTION
In survey sampling, one of the primary objectives is to estimate unknown population parameters such as the
population mean. To improve the accuracy of these estimates, researchers often make use of auxiliary
information related to the study variable. When auxiliary variables exhibit correlation with the variable of
interest, incorporating them into estimation procedures can significantly reduce sampling variability and increase
the precision of the resulting estimates. Traditional estimation methods such as the ratio, product, and regression
estimators are well-known techniques that exploit auxiliary information to produce more reliable population
estimates.
Despite these advantages, many real-world surveys encounter the problem of nonresponse. Nonresponse arises
when certain selected units in the sample do not provide the required information completely (unit nonresponse)
or in part (item nonresponse) for the study variable. If this issue is not properly addressed, it can introduce bias
and reduce the efficiency of estimators. One of the earliest approaches for handling nonresponse in survey
sampling was developed by Hansen and Hurwitz (1946), who proposed a subsampling strategy for
nonrespondents. Their method involves conducting follow-up surveys on a subset of the nonresponding units to
obtain the missing information. This technique has become an important foundation for many later developments
in survey sampling theory.
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Another important approach that improves estimation efficiency is double sampling or two phase sampling. This
design is particularly useful in situations where the study variable is difficult or costly to collect, whereas
auxiliary variables are easier to obtain. In a double sampling design, a large first-phase sample is used to collect
information on auxiliary variables. Subsequently, a smaller second-phase sample is selected from the first sample
to collect information on the study variable along with the auxiliary variables. The information obtained in the
first phase can then be used to construct improved estimators for the second phase, leading to greater estimation
precision and reduced survey costs.
To further enhance estimator performance, many researchers have proposed exponential ratio-type estimators
that incorporate auxiliary information through nonlinear transformations. For instance, Bahl and Tuteja (1991)
introduced an exponential ratio-type estimator for estimating the population mean under simple random
sampling. Their work stimulated numerous extensions that adapted exponential-type estimators to more complex
sampling situations. Singh et al. (2009) developed an exponential estimator that accounts for nonresponse in
both the study and auxiliary variables. In a related contribution, Ozel Kadilar (2016) proposed another
exponential estimator that effectively utilizes auxiliary information, while Ünal and Kadilar (2020) extended this
approach to situations involving nonresponse in survey data. Other studies have developed exponential ratio-
type estimators for population mean in the presence of nonresponse incorporating the use of auxiliary variables
to improve accuracy in double sampling (Hazra, 2015; Khan & Khan, 2022; Oguagbaka et al., 2024).
Although considerable progress has been made in developing estimators for survey data with nonresponse, there
is still a need for improved methods that effectively combine multiple auxiliary variables within a double
sampling framework. In some practical survey situations, more than one auxiliary variable is available and these
variables can provide valuable additional information that enhances the accuracy of population estimates.
However, relatively limited research has focused on exponential ratio-type estimators that simultaneously utilize
two auxiliary variables under double sampling in the presence of nonresponse.
Motivated by these gaps in the literature, this study proposes a new exponential ratio-type estimator using two
auxiliary variables under double sampling in the presence of nonresponse. The statistical properties of the
proposed estimator, including its bias and mean square error, are derived using first-order approximation
techniques. Furthermore, the efficiency of the estimator is evaluated by comparing the mean square error with
those of several existing estimators. Numerical illustrations are also presented to demonstrate the performance
of the proposed estimator under 5% nonresponse and varying subsampling rates
Double Sampling Framework
Consider a finite population consisting of N units from which information about a study variable y is required.
Let x and z denote two auxiliary variables that are correlated with the study variable. In many practical surveys,
the values of auxiliary variables are easier and less expensive to obtain compared to the study variable.
Under the double sampling (two-phase sampling) design, a large first-phase sample of size n' (where n' < N) is
selected from the population using simple random sampling without replacement. During this phase, information
on the auxiliary variables x and z is collected.
From this first-phase sample, a smaller second-phase sample of size n (n < n') is drawn. In the second phase,
observations are taken on the study variable y in addition to the auxiliary variables. The sample means obtained
from the first and second phases are then used to construct improved estimators for the population mean.
In survey investigations, it is common for some selected units to fail to provide the required information for the
study variable. This situation is referred to as nonresponse. When nonresponse occurs, the sample is typically
divided into two groups: respondents and nonrespondents.
Following the approach proposed by Hansen and Hurwitz (1946), a subsampling technique is applied to the
group of nonrespondents. Suppose that among the selected second-phase sample, n
1
units respond while n
2
units
do not respond. A subsample of size r is then drawn from the nonrespondents, and additional efforts are made to
obtain their responses.
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The information obtained from respondents and the subsampled nonrespondents is combined to form an unbiased
estimator of the population mean. This procedure helps to reduce the bias that may arise due to missing responses
in the survey.
Baseline Estimators
Bahl and Tuteja (1991) proposed an exponential ratio-type estimator for population mean in simple random
sampling as:
1
Singh et al (2009) proposed an exponential ratio-type estimator in the presence of non-response on both the
study and auxiliary variables by adopting the exponential type estimator in (1) as:
2
Ozel Kadilar (2016) proposed an exponential type estimator as:
3
Unal and Kadilar (2020) proposed an exponential estimator in the presence of non-response on both the study
and auxiliary variables by adopting the exponential type estimator in (3) as:
4
Abifade (2020) proposed an exponential ratio-type estimator in double sampling as:
5
The Proposed Estimator
Motivated by the work of Singh et al (2009) and Unal and Kadilar (2020), we adopt the estimator in (5) and
hereby proposed an exponential ratio-type estimator for estimating population mean using two auxiliary
variables when nonresponse occurs on the study variable only as:
6
Where
nonresponse
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To obtain the Bias and Mean Square Error (MSE) of the proposed estimator
, we define the relative error
terms and their expectations as:
Let
,
,
,
,
And
,
,
,
,
Such that
And
Where
,
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The estimator
can be expressed in terms of
’s as follows:
7
Assuming that
then we expands the right hand side of equation 7
to the second degree of approximation using negative binomial series, we have
8
9
Rewriting equation 9 to second degree of approximation, we have
Taking the exponential of the R.H.S. and expand, we have
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10
Equation 10 to second degree approximation, we have
11
Subtracting
from both sides, we have
12
Taking the expectation of both sides of equation 12, we have the bias of
to the first degree approximation
as
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Estimator
is approximately unbiased if the value of the constant is
To obtain the error function of the estimator, we re-write equation 12 to first degree approximation, we have
13
Squaring both sides of equation 13 and neglecting terms of ‘s involving power greater than two, we have
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14
Taking the expectation of both sides of equation 14, we get the MSE of the estimator
to the first degree
approximation, we have
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15
Differentiating equation 15 w.r.t. , we have
16
Substitute
in equation 3.16, we have
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Let A =
And B =
Hence,
17
Some Existing Estimators with their MSE
Hanson and Hurwitz 1946 proposed an unbiased estimator for population mean as:
18
and its MSE as:
19
Singh et al 2009 proposed an exponential estimator for estimation of mean in the presence of nonresponse as:
20
and its MSE as:
21
Hazra 2015 proposed an exponential ratio-type estimator for estimating population mean using auxiliary variable
with double sampling in the presence of nonresponse as:
22
and its MSE as:
23
Khan and Khan 2022 proposed exponential ratio-type estimators of population mean using two auxiliary
variables under nonresponse as:
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24
and its MSE as:
25
Oguagbaka et al 2024 proposed a ratio estimator for double sampling procedure with nonresponse as:
26
and its MSE as:
27
Efficiency Comparison
Some conditions were identified under which the proposed estimator performs better than the existing estimators
in terms of efficiency by comparing the mean square errors of the estimators.
Efficiency Comparison of the proposed estimator
with Oguagbaka (2024) estimator
The proposed estimator
will be more efficient than the Oguagbaka et al (2024) estimator if and only if
28
Equation 28 implies
which also implies
29
Substituting equation 27 and 17 in equation 29, we have
Hence, the proposed estimator
will be more efficient than the Oguagbaka et al (2024) estimator if
30
Efficiency Comparison of the proposed estimator
with Khan and Khan (2022) Estimator
The proposed estimator
will be more efficient than the Khan and Khan (2022) estimator if and only if
31
Equation 31 implies
which also implies
32
Substituting equation 25 and 17 in equation 32, we have
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Hence, the proposed estimator
will be more efficient than the Khan and Khan (2022) estimator if
33
Efficiency Comparison of the proposed estimator
with Hazra (2015) estimator
The proposed estimator
will be more efficient than the Hazra (2015) estimator if and only if
34
Equation 34 implies
which also implies
35
Substituting equation 23 and 17 in equation 35, we have
Hence, the proposed estimator
will be more efficient than the Hazra (2015) estimator if
36
Efficiency Comparison of the proposed estimator
with Singh (2009) estimator
The proposed estimator
will be more efficient than the Singh (2009) estimator if and only if
37
Equation 37 implies
which also implies
38
Substituting equation 21 and 17 in equation 38, we have
Hence, the proposed estimator
will be more efficient than the Singh (2009) estimator if
39
Efficiency Comparison of the proposed estimator
with Hansen and Hurwitz (1946) Estimator
The proposed estimator
will be more efficient than the Hansen and Hurwitz (1946) estimator if and only
if
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40
Equation 40 implies
which also implies
41
Substituting equation 19 and 17 in equation 41, we have
Hence, the proposed estimator
will be more efficient than the Hansen and Hurwitz (1946) estimator if
42
RESULT AND DISCUSSION
This study uses a real life data obtained from the payment voucher of the Akure South Local Government,
Nigeria consisting of 1,023 workers. The double sampling uses tax as the study variable y, grade level as the
auxiliary variable x and gross payment as auxiliary variable z. The double sampling is done with the first sample,
second sample and subsample sizes at four levels (k = 2, 3, 4, 5) at 5% nonresponse level.
The data obtained was analyzed using SPSS, Scilab software and Microsoft Excel.
Table 1Sample Estimates at 5% Nonresponse
Term
k = 2
k = 3
k = 4
k = 5
972
972
972
972
51
51
51
51
600
600
600
600
450
450
450
450
427
427
427
427
23
23
23
23
0.43988
0.43988
0.43988
0.43988
0.58651
0.58651
0.58651
0.58651
0.9501
0.9501
0.9501
0.9501
0.0499
0.0499
0.0499
0.0499
0.9489
0.9489
0.9489
0.9489
0.0511
0.0511
0.0511
0.0511
0.001245
0.001245
0.001245
0.001245
0.000689
0.000689
0.000689
0.000689
0.0001109
0.0002218
0.0003327
0.0004436
5528.6505
5528.6505
5528.6505
5528.6505
9.03
9.03
9.03
9.03
87114.2835
87114.2835
87114.2835
87114.2835
9.12
9.12
9.12
9.12
89272.9517
89272.9517
89272.9517
89272.9517
7176.68099
7176.68099
7176.68099
7176.68099
3.259
3.259
3.259
3.259
63589.33128
63589.33128
63589.33128
63589.33128
51504750.06
51504750.06
51504750.06
51504750.06
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10.624
10.624
10.624
10.624
4043603052
4043603052
4043603052
4043603052
15958.438
15958.438
15958.438
15958.438
450727541.1
450727541.1
450727541.1
450727541.1
150959.333
150959.333
150959.333
150959.333
0.682
0.682
0.682
0.682
0.988
0.988
0.988
0.988
0.728
0.728
0.728
0.728
1.298089
1.298089
1.298089
1.298089
0.361027
0.361027
0.361027
0.361027
0.729953
0.729953
0.729953
0.729953
1.685035
1.685035
1.685035
1.685035
0.130341
0.130341
0.130341
0.130341
0.532831
0.532831
0.532831
0.532831
2.452986
2.452986
2.452986
2.452986
1.756369
1.756369
1.756369
1.756369
0.360234
0.360234
0.360234
0.360234
1502.1120
1502.1120
1502.1120
1502.1120
12
8
6
5
0.787
0.792
0.759
0.776
0.994
0.974
0.980
0.999
0.815
0.838
0.684
0.799
1.294723
1.049499
0.749364
0.972256
0.379059
0.333809
0.361265
0.302917
0.902711
0.692807
0.523851
0.669536
1.676307
1.1014473
0.561546
0.945282
0.143685
0.111429
0.130512
0.091759
0.814887
0.479982
0.274420
0.448278
2.687352
2.489638
1.574968
2.490271
1.426058
1.4748340
1.401865
1.450971
0.342423
0.4039821
0.471519
0.361389
11277.57472
9587.60524
3217.92982
3494.80149
3.696
3.338
3.312
2.302
106364.2757
86953.30703
42814.00917
50007.29158
127183691.6
91922174.19
10355072.30
12213637.48
13.659
11.143
10.967
5.30
11313359145
756087.7603
1833039381
2500729211
32792.965
25343.180
8091.307
6242.389
1192671034
811655078.0
135015484.8
174625578.6
320562.136
243366.351
96940.208
91959.364
*
8710.4175
9135.4150
4294.2150
3594.5280
9.75
10.00
9.17
7.60
117827.6158
125508.6888
81729.3917
74689.4740
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Table 2 Mean Square Error (MSE) at 5% Nonresponse
Estimator
k=2
k=3
k=4
k=5
Y
FA1
54601.35
56482.19
58774.61
61902.44
Y
HH
75916.48
78304.12
80892.77
83746.55
Y
Set
61283.44
63152.76
65308.41
67819.22
Y
H
72648.31
74902.56
77381.94
79942.87
Y
KK
58463.72
60894.36
63811.49
67295.18
Y
Oget
61702.48
63594.21
66104.73
68337.90
Figure 1 Bar chart showing the Mean Square Error at 5% Nonresponse
Table 2 and Figure 1 show that at 5% nonresponse, the MSE of all estimators increases as the value of k increases.
The proposed estimators, Y
FA1
achieves the minimum MSE at all levels of k respectively.
Table 3 Coefficient of Variation (CV) at 5% Nonresponse
Estimator
k=2
k=3
k=4
k=5
Y
FA1
2.68
2.60
5.64
6.93
Y
HH
3.16
3.06
6.63
8.03
Y
Set
2.84
2.74
5.94
7.20
Y
H
3.09
2.99
6.48
7.83
Y
KK
2.76
2.69
5.83
7.09
Y
Oget
2.85
2.75
5.98
7.26
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
YFA1 YHH YSet YH YKK YOget
MSE
Estimators
k=2
k=3
k=4
k=5
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Figure 2Bar chart showing the Coefficient of Variation at 5% nonresponse
Table 3 and figure 2 show that the CV increases as k increases and it was observed that the CV values are
generally low across all estimators, however, the proposed estimator outperform the existing ones.
Table 4 Relative Efficiency (RE) at 5% Nonresponse
Estimator
k=2
k=3
k=4
k=5
Y
FA1
139.04
138.64
137.63
135.29
Y
HH
100
100
100
100
Y
Set
123.88
123.99
123.86
123.48
Y
H
104.50
104.54
104.54
104.76
Y
KK
129.85
128.59
126.77
124.45
Y
Oget
123.04
123.13
122.37
122.55
Figure 3 Graph showing the Relative Efficiency at 5% Nonresponse
0
1
2
3
4
5
6
7
8
9
YFA1 YHH YSet YH YKK YOget
Coefficient of variation
Estimators
k=2
k=3
k=4
k=5
0
100
200
300
400
500
600
YFA1 YHH YSet YH YKK YOget
RE
Estimators
k=5
k=4
k=3
k=2
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Table 4 and figure 3 show that the proposed estimator had the highest RE value across all subsampling rate k
which implies that the proposed estimator is more efficient than other estimators.
DISCUSSION
The performance of the proposed estimator was evaluated using numerical illustrations and compared with
several existing estimators, including estimators by Hansen–Hurwitz, Singh et al., Hazra, Khan and Khan, and
Oguagbaka et al.
Summarily, the mean square error (MSE) values obtained from the numerical study demonstrate that the
proposed estimator consistently produces smaller MSE values, lower CV values than competing estimators
across different values of (k).
In conclusion, the theoretical and empirical results demonstrate that the proposed estimator provides a more
efficient alternative for estimating the population mean in double sampling in the presence of nonresponse. The
reduction in MSE can be attributed to the effective use of two auxiliary variables within an exponential
framework, which helps capture additional information about the population and reduce estimation variability.
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9. Särndal, C. E., Swensson, B., and Wretman, J. (2003). Model Assisted Survey Sampling. Springer.
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